Physical Chemistry Third Edition

22 1 The Behavior of Gases and Liquids

c.Find the pressure of 1.000 mol of nitrogen at a volume of 1.000 L and a temperature of

298.15 K using the van der Waals equation of state. Find the pressure of an ideal gas under

the same conditions.

Another common equation of state is thevirial equation of state:

PVm

RT

 1 +

B 2

Vm

+

B 3

Vm^2

+

B 4

Vm^3

+ ··· (1.3-3)

which is a power series in the independent variable 1/Vm. TheBcoefficients are called

virial coefficients. The first virial coefficient,B 1 , is equal to unity. The other virial

coefficients must depend on temperature in order to provide an adequate representation.

Table A.4 gives values of the second virial coefficient for several gases at several

temperatures.

An equation of state that is a power series inPis called thepressure virial equation

of state:

PVmRT+A 2 P+A 3 P^2 +A 4 P^3 + ··· (1.3-4)

The coefficientsA 2 ,A 3 , etc., are calledpressure virial coefficientsand also must depend

on the temperature. It can be shown thatA 2 andB 2 are equal.

EXAMPLE 1.9

Show thatA 2 B 2.

Solution

We solve Eq. (1.3-3) forPand substituting this expression for eachPin Eq. (1.3-4).

P

RT

Vm

+

RT B 2

Vm^2

+

RT B 3

Vm^3

+ ···

We substitute this expression into the left-hand side of Eq. (1.3-4).

PVmRT+

RT B 2

Vm

+

RT B 3

Vm^2

+ ···

We substitute this expression into the second term on the right-hand side of Eq. (1.3-4).

PVmRT+A 2

RT

Vm

+

RT B 2

Vm^2

+

RT B 3

Vm^3

+ ···

If two power series in the same variable are equal to each other for all values of the variable,

the coefficients of the terms of the same power of the variable must be equal to each other.

We equate the coefficients of the 1/Vmterms and obtain the desired result:

A 2 B 2

Exercise 1.8

Show thatA 3 

1

RT

(

B 3 −B^22

)

.